I have three problems when fitting panel regression model with fixed effect using xtreg,fe

first problem
I fittted two models.
first model includes year fixed effect and region fixed effect and id fixed effect.
second model includes year fixed effect, region fixed effect.
the problem is, the model that I added id fixed effect has low R-sq. I thought when I add more variable (and also fixed effect), r-sq increases.
Am I right? or the number of fixed effect is not associated with value of r-square?
I am interested in the relationship between Y(logfood) and uet, pet. the other variables are added as the control variables.
here is the output
xtset pid year, yearly

first model
xtreg logfood uet2_lam1_0329 pet1_lam1_0329 iliquid_w_i liquid_w_i loginc age cur_empl size sex edu maritial i.region i.year ,fe
note: sex omitted because of collinearity
note: 2017.year omitted because of collinearity

Fixed-effects (within) regression Number of obs = 18265
Group variable: pid Number of groups = 3575

R-sq: within = 0.1340 Obs per group: min = 1
between = 0.0648 avg = 5.1
overall = 0.0823 max = 15

F(39,14651) = 58.12
corr(u_i, Xb) = -0.2528 Prob > F = 0.0000

--------------------------------------------------------------------------------
logfood | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------------+----------------------------------------------------------------
uet2_lam1_0329 | -5.331916 2.165693 -2.46 0.014 -9.576947 -1.086884
pet1_lam1_0329 | -.0555771 .0240872 -2.31 0.021 -.102791 -.0083632

Second model
xtreg logfood uet2_lam1_0329 pet1_lam1_0329 iliquid_w_i liquid_w_i loginc age cur_empl size sex edu maritial i.region ,fe i(year)
warning: existing panel variable is not year

Fixed-effects (within) regression Number of obs = 18265
Group variable: year Number of groups = 15

R-sq: within = 0.4786 Obs per group: min = 868
between = 0.1105 avg = 1217.7
overall = 0.4698 max = 1421

F(27,18223) = 619.55
corr(u_i, Xb) = 0.0048 Prob > F = 0.0000

--------------------------------------------------------------------------------
logfood | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------------+----------------------------------------------------------------
uet2_lam1_0329 | -6.87497 1.522391 -4.52 0.000 -9.859 -3.890939
pet1_lam1_0329 | -.0526164 .0195237 -2.69 0.007 -.0908847 -.014348



Second problem
when I report the value of R-square on my paper. which r-square do I have to use?
I am interested in the relationship between Y(logfood) and uet, pet. the other variables are added as the control variables.
what is the meaning and value of r-sq on panel regression?
and if I want to compare r-square with pooled ols r-square, How can I do this?


Third problem
When I fitted the same model on the first problem, I want to cluster standard deviation by pid on both models.
I operated following codes. the problem is that I cannot cluster second model by pid using xtreg. because panels are not nested.
So I use reg command, clustering by pid and I get same coefficients. but I cannot get same kinds of r-square that xtreg command reports.
Is there any way to report consistent r-square values on reg and xtreg?

first model clustering by pid
. xtreg logfood uet2_lam1_0329 pet1_lam1_0329 iliquid_w_i liquid_w_i loginc age cur_empl size sex edu maritial i.region i.year ,fe cluster
> (pid)
note: sex omitted because of collinearity
note: 2017.year omitted because of collinearity

Fixed-effects (within) regression Number of obs = 18265
Group variable: pid Number of groups = 3575

R-sq: within = 0.1340 Obs per group: min = 1
between = 0.0648 avg = 5.1
overall = 0.0823 max = 15

F(38,3574) = .
corr(u_i, Xb) = -0.2528 Prob > F = .

(Std. Err. adjusted for 3575 clusters in pid)
--------------------------------------------------------------------------------
| Robust
logfood | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------------+----------------------------------------------------------------
uet2_lam1_0329 | -5.331916 2.544247 -2.10 0.036 -10.32024 -.3435932
pet1_lam1_0329 | -.0555771 .0298144 -1.86 0.062 -.114032 .0028778


Second model clustering by pid using xtreg
. xtreg logfood uet2_lam1_0329 pet1_lam1_0329 iliquid_w_i liquid_w_i loginc age cur_empl size sex edu maritial i.region ,fe i(year) cluste
> r(pid)
warning: existing panel variable is not year
panels are not nested within clusters


Second model clustering by pid using reg
xi : reg logcon uet2_lam1_0329 pet1_lam1_0329 iliquid_w_i liquid_w_i loginc age cur_empl size sex edu maritial i.year i.region , vce(cl p
> id)
i.year _Iyear_1998-2018 (naturally coded; _Iyear_1998 omitted)
i.region _Iregion_1-19 (naturally coded; _Iregion_1 omitted)
note: _Iyear_1999 omitted because of collinearity
note: _Iyear_2000 omitted because of collinearity
note: _Iyear_2001 omitted because of collinearity
note: _Iyear_2002 omitted because of collinearity
note: _Iyear_2003 omitted because of collinearity
note: _Iyear_2018 omitted because of collinearity

Linear regression Number of obs = 18292
F( 40, 3578) = .
Prob > F = .
R-squared = 0.7768
Root MSE = .35504

(Std. Err. adjusted for 3579 clusters in pid)
--------------------------------------------------------------------------------
| Robust
logcon | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------------+----------------------------------------------------------------
uet2_lam1_0329 | -6.387408 1.901937 -3.36 0.001 -10.1164 -2.658419
pet1_lam1_0329 | -.1468916 .019314 -7.61 0.000 -.1847591 -.109024